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Nat. Hazards Earth Syst. Sci. Discuss.,
1, 1857–1893,
Natural
Hazards2013
www.nat-hazards-earth-syst-sci-discuss.net/1/1857/2013/
and Earth System
doi:10.5194/nhessd-1-1857-2013
Sciences
© Author(s) 2013. CC Attribution 3.0 License.
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Model Development1857
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Correspondence to: L. C. Wu ([email protected])
|
Earth System
Received: 28 March 2013 – Accepted:
26 April 2013 – Published: 14 May Dynamics
2013
Dynamics
Open Access
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School of Navigation, Wuhan University of Technology, Wuhan, China
2
Hubei Inland Shipping Technology Key Laboratory, Wuhan, China Earth System
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Climate1,2
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Wu
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1, 1857–1893, 2013
Safe-economical
route and its
assessment model
L. C. Wu et al.
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Safe-economical route and its
assessment model of a ship to avoid
Biogeosciences
Biogeosciences
tropical cyclones
using dynamic
forecast
environment
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Measurement
Techniques
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In heavy sea conditions related to tropical cyclones (TCs), losses to shipping caused
by capsizing are greater than other kinds of accidents. Therefore, it is important to
consider capsizing risk in the algorithms used to generate safe-economic routes that
avoid tropical cyclones (RATC). A safe-economic routing and assessment model for
RATC, based on a dynamic forecasting environment, is presented in this paper. In
the proposed model, a ship’s risk is quantified using its capsizing probability caused
by heavy wave conditions. Forecasting errors in the numerical models are considered
according to their distribution characteristics. A case study shows that: the economic
cost of RATCs is associated not only to the ship’s speed and the acceptable risk level,
but also to the ship’s wind and wave resistance. Case study results demonstrate that
the optimal routes obtained from the model proposed in this paper are significantly
superior to those produced by traditional methods.
Discussion Paper
Abstract
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Weather hazards are the main threat to shipping. The goal of weather routing is to plan
routes that avoid weather hazards safely and economically. Tropical Cyclones (TCs),
as a kind of hazardous weather, cause extensive damage to the ship and crew. The
total loss caused by ship capsizing is very serious to the ship company and cargo
owner. Compared to the other accidents, the loss of cargo and ship damage caused
by other accidents is much less in TCs. Ships can avoid TCs safely with routes based
on the methods applied in navigation practice – Sector diagram typhoon avoidance
method (Chen, 2004); 34KT rule (Holweg, 2000); and the Diagram of the 1-2-3 rule
(Wisniewski et al., 2009) – but routes generated by these methods ignore costs and
increase shipping expenses. Furthermore, in these models the ship’s performance in
resisting wind and waves is not considered.
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1 Introduction
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2 Literature review
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The resolution precision of ocean and atmospheric models are increasing in tandem
with the rapid development of computational power, thus allowing more possibilities
for guiding ship routing with precise and less uncertain forecast results (Delitala et al.,
2010). Many researchers have done weather routing work using ocean and weather
numerical models, but the forecast errors are not considered in their work (Padhy et al.,
2008; Maki et al., 2011). Nowadays, it is feasible to find routes to avoid TCs (RATC)
using dynamic wind and wave fields as forecasted by numerical models. A minimum
economic cost route was designed to avoid TCs from the following aspects: (1) the
forecast errors of models are considered in the route design; (2) the ship’s characteristics are considered in assessing the risk in the heavy weather conditions; (3) the ship’s
risk is quantified; (4) the ship’s speed loss is considered.
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The relevant literature includes research on weather routing, numerical forecast error,
RATC, and vessel risk analysis.
There are several widely used methods for weather routing. These methods
are based on the isochrone method and have been refined since its introduction.
James (1957) proposed an isochrone method, which was widely used for decades.
Based on the work of James, improvements have been made to update the method
(Hagiwara, 1989). Chen (1978) developed a dynamic programming algorithm to solve
the minimum voyage cost problem under uncertainty constraints. In the algorithm, the
sea-keeping features of the ship are a function of weather. McCord et al. (1999) investigated the potential for strategic ship routing through dynamic currents to determine
3 day routes. The results showed that this kind of strategic ship routing can reduce
fuel consumption by 25 % on average. Delitala showed that weather routing improves
ship performance by 37 % thus supporting ship captains throughout an entire voyage
(Delitala et al., 2010). Panigrahi et al. (2008) and Padhy et al. (2008) optimized the
ship’s route based on WAM (Wave Modelling model). Maki et al. (2011) designed
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a real-coded genetic algorithm weather routing method to calculate the safety ratio
and fuel efficiency based on the probability of accidents caused by parametric rolling.
Soda et al(2011) gathered and used high-resolution wind and wave data forecasted
using SWAN (Simulating Waves Nearshore model) and WRF (Weather Research and
Forecasting Model) to study wave and wind effects on ship’s manoeuvring.
The weather routing problem is complicated since weather conditions are uncertain.
The forecast error in numerical models must be considered when designing weather
routing (Magirou and Psaraftis, 1992). Hopkins examined the offshore forecast statistical errors derived from the numerical forecasting models. RMS (root mean square)
errors in wind speed and wave height forecasts have a seasonal variation (Hopkins,
1997). The RMS errors of the WaveWatchIII model against altimeter and buoy data are
15 % of the mean observed wave heights for most of the global domain (Tolman, 2002).
Bedard (2008) evaluated the forecast technology of WaveWatchIII based on the comparison with buoy-measured wave data in deep water of Washington and Oregon, USA
Chu et al. compared the model result of WaveWatchIII with the T/P (Topex/Poseidon)
significant wave height (SWH) data over the satellite crossover points in the South
China Sea (SCS). The results showed that the model errors of SWH had Gaussiantype distribution (2004). The research of Guo and Hou (2010) coincides with Chu’s
(Chu et al., 2004) research results in that the WaveWatchIII forecast error of SWH
follows a Gaussian distribution.
Although the literature on weather routing is rich, little work focuses on RATC. The
sector diagram typhoon avoidance method (Chen, 2004), and the 34KT rule (Holweg,
2000) are widely used in navigation. Wisniewski discussed the application of the 1-2-3
rule for calculations for routing vessels using evolutionary algorithms (2009). Liu(2006)
built a safety-economic decision-making model of RATC using risk theory and fuzzy information optimization. Zhang (2010) proposed a multilevel decision method for RATC
using multi-source track forecasting. Wu (2010) built a model to evaluate the benefit for
RATC. Wisniewski and Kaczmarek (2012) analysed reactions, such as reducing speed
and changing course, when a ship is avoiding a TC. At present, these RATC methods
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are mainly based on TC forecast tracking, experience, and fuzzy analysis. All these
methods, in the final instance rely on qualitative judgement. These methods also do
not consider the performance or reactions of different vessels under the same wave or
wind conditions. Therefore, different vessels might require different routing strategies
to safely and economically avoid TC.
The quantification of risk to ships under heavy weather conditions is an important
problem for weather routing. Most researchers assess a ship’s risk using fuzzy analysis, the risk cannot be quantified (Zhang et al., 2010; Liu et al., 2006). In heavy sea
conditions related to tropical cyclones (TCs), losses to shipping caused by capsizing
are the total losses of ship and cargo, which are much greater than other kinds of accidents. So, it is reasonable to considered the capsizing probability as the risk level
in RATC. In recent years, capsizing probability has received attention as an empirical
measure to quantify the ship’s risk. Shen and Huang(2000) studied the length of time
before capsizing and the capsizing probability of ship based on Markov chain theory.
Huang (2001) studied the capsizing probability of a ship under the combined action of
beam wind and beam sea. In this method, the capsizing probability of every random
heeling in an unstable domain is calculated according to the density of heeling extreme
value. Thompson (Thompson, 1990; Thompson et al., 1992) used the theory of safe
basis erosion to study the ship capsizing probability. Shi et al. (2011) studied the calculation method of ship’s movement and capsizing probability in random waves and
winds using formula of Gauss-Legendre based on the path integration techniques. Gu
(2006) calculated the ship’s rolling probability using a new path integration method,
which avoided the problem of solving the equations of Fokker–Planck–Kolmogorov
(FPK). The method of Melnikov is also used to calculate the ship’s capsizing probability (Falzarano et al., 1992; Bikdashi et al., 1994; Tang et al., 2004).
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10
(1)
In which, t 0 = t − ∆t, E is the external environment at location x 0 at time t 0 . The ship is
controlled by U during time ∆t. The ship will arrive at location x at time t. The economic
cost C of the whole voyage can be expressed as:
(2)
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In which, Coil is the fuel consumption per unit time, and related to ship’s speed, displacement and the oil price; Ct is ship’s profitability per unit time, and related to type of
ship and the production plan.
How to improve the economic benefit of a ship based on safety criteria is the concept behind of RATC. An economical and safe RATC based on a dynamic forecasting
environment increases ship safety and reduces costs using the wave and wind condition forecasted by numerical models. Costs may be very different given the same risk
because of the difference between the ship and cargo’s value. To assess the economic
benefit of RATC, a benefit-cost ratio for a route is built. A ship’s safety can be valued
as the safety probability multiplied by the ship’s fixed assets including the value of both
Discussion Paper
Coil (D, V , Qa ) + Ct
T0
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C(xD , t) =
ZT
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(x, t) = f (x 0 , t 0 , E , U, M)
|
The movement of ships sailing on the ocean can be described as the change of ship’s
position over time. The ship’s position (x(lon,lat)) and time (t) form the ship’s track.
The ship’s dynamic response to varying environmental conditions can be described by
a control vector U and restraint vector M. U describes the ship’s heading direction and
speed. M are restraint conditions. E describes the external environment which varies
with time. The dynamic process of a ship can be expressed as:
Discussion Paper
3 Mathematic model
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Ra =
(1 − pi −1,i )
(4)
i =1
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Psafe =
n
Y
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20
Where, Ccost is the economic cost of the whole voyage, including the time cost and
fuel consumption. Psafe is the safety probability of ship, considered as the probability
that ship will not capsize. Mall is the value of the ship and the cargo. Ra is the total
cost of ensuring a ship’s economic safety in economic cost units when avoiding TCs.
Therefore, the smaller the Ra is, the less it takes to ensure a ship’s safety when avoiding
TCs.
The mathematical model for RATC can be considered as a ship’s avoidance of an
obstruction that changes shape and position over time. The accident probability of ship
must not exceed a acceptable risk level for the operator. Assuming that a ship begins to
take an action to avoid TC at time t0 , and will pass through the TC after time T . Then,
during time T , the ship’s speed and course will change predetermined route. The points
at which the ship’s speed or course is changed are considered to be waypoints. The
avoidance process is regarded as complete when the ship arrives at the next waypoint
of the original predetermined route. The whole process of avoiding TCs therefore, can
be regarded as an optimal path problem.
Make x0 as the starting point for ships to avoid TC. xn (n = 1, 2, . . .) are the alternative
waypoints of the route; li ,i +1 is a segment of the route between two adjacent alternative
waypoints; di ,i +1 is the distance between any two adjacent alternative waypoints; pi ,i +1
is the accidental probability of a ship in a unit time between any two adjacent alternative
waypoints; vi ,i +1 is the average speed-to-ground when ship is sailing on segment li ,i +1 .
Then, the ship’s safety probability Psafe in the whole process is:
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Mall · Psafe
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(3)
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Ccost
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ship and cargo. Then, the benefit-cost ratio (Ra) of RATC can be expressed as:
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Ccost =
n
X
di −1,i
i =1
vi −1,i
· (Ct + Coil )
4 The RATC algorithm
4.1 Relevant parameters
Discussion Paper
The acceptable risk level restriction is: pi −1,i < pa .
In which, pa is the acceptable accident probability of ship capsizing;
i is an alternative waypoint of the route;
n is the amount of the alternative waypoints;
Qoil ∝ D 2/3 · V · Qa is oil cost in per unit time;
D is displacement of the ship, and Qa is the oil price.
The most economical route (the minimum economical cost route) based on the safety
is min{Ccost }.
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15
Waves are stimulated by the wind during TCs. Wind and waves are strongly correlated.
In this paper, the risk factor is simplified using the risk of capsizing in random waves.
The differential equation of ship’s nonlinear rolling motion in random wave conditions
is:
20
(7)
|
(I + I1 (ω))φ̈ + B1 (ω)φ̇ + B2 (ω)φ̇3 + ∆GZ = Fsea (τ)
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4.1.1 Ship capsizing probability
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(6)
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The benefit-cost ratio (Ra) of the route is :
Pn di −1,i
i =1 vi −1,i · (Ct + Coil )
Ra =
Q Mall · ni=1 (1 − pi −1,i )
(5)
Discussion Paper
The economic cost in the whole process is:
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Where, I is the rotational moment of inertia around an assumed rolling centre; I1 (ω) is
the added moment of inertia due to the ambient fluid; B1 and B2 are linear and cubic
damping coefficients respectively; ∆ is the ship displacement; GZ is the righting arm of
a rolling ship; Fsea (τ) is the external excitation resulting from random beam seas, the
over-dots denote differentiation with respect to time τ; ω is the wave frequency.
The righting arm is approximated by the following odd cubic polynomial of φ,
|
(8)
The excitation moment resulting from the random seas is expressed as:
√
Fsea =
I0 α0 ω20
N
p
2$ X
(n$)2 S cos(ωn t + ξn )
g
(9)
n=1
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15
In which, I0 = I +I1 (ω); ω0 is the natural frequency of ship’s roll; $ is the interval of wave
frequency; S is the excitation intensity of white noise; ξn is the random phase angle in
(0, 2π).
When a ship is at sea, the wave encounter angle (χ ) is the angle of wave encounter
between the heading direction and wave direction as illustrated in Fig. 1. Also shown
in Fig. 1 are the ship’s breadth, B; the ship’s speed, U; the wave’s length, λ; and the
frequency of wave encounter, ωe .
The encounter frequency between wave and ship is:
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Safe-economical
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GZ(φ) = C1 φ − C3 φ3
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(10)
The encounter spectrum’s relationship with the wave spectrum:
20
Se (ω) =
S(ω)
1 − 2ω/g µ cos χ
(11)
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ω
ωe = ω −
µ cos χ
g
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Fsea = I0 α0 ω20 π sin χ
N
X
n=1
hn
cos(ωen t + ξn )
λn
(13)
p
in which wave height hn = 2 2$S (n$) and wave length λn =
.
ẍ(t) + εδ1 ẋ(t) + εδ2 ẋ 3 (t) + x(t) − αx 3 (t) = εf (t)
In which, x = φ; t = ω0 τ; εδ1 =
α=
Ω=
ω
ω0 ;
ω0 =
q
∆C1
;
I+I1 (ω)
f (t) =
ε is a very small value; the time t is controlled by the natural frequency; εδ1
and εδ2 are linear and nonlinear dimensionless dampers, respectively.
The joint probability density of φ and φ̇ can be solved by solving the corresponding Fokker–Planck–Kolmogorov (FPK) equation for Eq. (14). In this paper the method
proposed by Gu (2006) is used to solve the FPK equation. The joint probability is:
(
!)
(
!)
2
x14
x24
x22
φ x1
1
PF x1 , x2 |φ = a exp −
−α
exp −
(15)
δ1 + δ2
2
4
2
4
D
D
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εδ2 =
C3
;
C1
|
Fsea
;
∆C1
B2 (ω)
ω ;
I+I1 (ω) 0
(14)
Discussion Paper
After dimensionless treatment, the differential equation for a ship’s rolling motion in
random seas, in which the white noise is considered, is as following,
B1 (ω)
ω0 ;
∆C1
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2πg
(n$)2
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2
Where g is gravitational acceleration; A = 8.10 × 10−3 g2 ; B = 3.11/H1/3
; H1/3 is the
SWH.
Therefore, the wave excitation torque for random waves is calculated using the following formula:
|
5
(12)
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The wave spectrum, with single parameter, as specified by the ITTC was used:
A
B
S(ω) =
exp −
ω5
ω4
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R∞
0
φ=
x2
3
δ1 x2 + δ2 x2
2
4
x2
x2
1
dx2
exp − D δ1 2 + δ2 4
R∞ 2
x24
x22
1
dx2
0 x2 exp − D δ1 2 + δ2 4
(16)
Discussion Paper
in which,
|
+∞
Z +∞
Z
PF x1 , x2 |φ dx1 dx2 = 1
(17)
−∞ −∞
5
10
−∞
15
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Ship speed loss in waves has an important effect on the estimate of the ship’s location
in the design of RATC. If the ship’s speed loss is not considered, it will cause error
when calculating the ship’s position over time. At present, there are many empirical
formulas to estimate the ship speed loss due to waves (Aertssen, 1969; James, 1957;
1867
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4.1.2 Ship speed loss in waves
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In which, φv1 is the angle of disappearing positive stability; φv2 is the angle of disappearing negative stability; P (x1 , t) marginal probability density.
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φv1
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D is the amplitude of excitation.
Then, marginal probability density of φ can be calculated based on the joint probability density. Thus, the ship capsizing probability P can be calculated using the following
formula:


φ
+∞
v2


Z
Z


P = max
P (x1 , t) dx1 ,
P (x1 , t) dx1
(18)




Discussion Paper
α is normalized parameter which can be calculated using the following formula:
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∆V
=h
V
0.45 ·
B
T
12 · · Cw
i ·%
L
L 2
+ 0.35 · 100 · L
100
(19)
Discussion Paper
Li et al., 2011). In this paper, however, the formula proposed by He and Dong (2009) is
used to estimate the ship speed loss in the heavy conditions. The formulas described
as follow:
|
10
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Although model forecasting accuracy is increasing, the forecast results from numerical
models still have errors caused by inaccurate initial and boundary conditions. If forecasting errors are not considered in designing RATC, the calculation of ship’s risk will
have significant errors. Dealing appropriately with the forecasting errors in numerical
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4.1.3 Forecast error of numerical model
20
Discussion Paper
H1/3 is the SWH, unit: m;
T1 is the wave feature period, unit: s.
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Safe-economical
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In which,
∆V is ship’s speed loss in waves, unit: kn;
V is ship’s designing speed, unit: kn;
L is length of two makefasts, unit: m;
B is the ship’s breadth, unit: m;
T is the ship’s draft, unit: m;
2
Cw is the index of wave grade, which can be calculated using Cw = K · T1 · H1/3
K is the correction factor, which can be calculated using the following formula.

L ≤ 150

−0.05 · H1/3 + 0.9
150 < L < 200
K = 1.3
(20)

0.0125 · H 2 + 0.05 · H + 0.7375 L ≥ 200
1/3
1/3
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(21)
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The positioning of the alternative waypoints is important to RATC design. Finding analytic solutions for the minimum economic cost of RATC, however, is intractable since
the alternative waypoints can be at any point in the sea. To reduce the computational
budget, the alternative waypoints must be artificially restricted by the area of heavy
seas and rough weather caused by TCs. In turn, if the interval of adjacent alternative
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4.1.4 Alternative waypoints
|
8
X
∆ri · ∆pi
P8
i =1
i =1 ∆ri
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p=
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models is important for weather routing. The SWH errors are important when calculating a ship capsizing probability.
In the research of Chu (Chu et al., 2004), the SWH errors from WaveWatchIII have
a Gaussian-type distribution with a small mean (µ) value of 0.02 m, as compared with
the T/P altimeter data in the SCS. The RMS error (σ) and correlation coefficient between the modelled (Hm ) and observed (Ho ) SWH are 0.48 m and 0.90, respectively. In
this research, 1330 samples were used for statistical analysis. The distribution of the
errors is shown in Fig. 2, where ∆H = Hm − Ho .
Taking the error range into consideration, a ship capsizing probability for the forecast
wave height is calculated based on the distribution of the wave model forecast errors.
In this paper, the error range was processed using the truncation method. Because the
probability of errors bigger than 2 m or less than −2 m is only 0.019, this paper only
considers the error range in [−2,2]. The division of the error range ∆Ei is shown in
Table 1, and the probability for each error range is ∆ri . According to the forecast value
of the wave height and the error range, the ship capsizing probability ∆pi in each error
range can be calculated. If the value that the forecast value adds to the error range is
less than 0, then the probability in this error range will be 0, because the wave height
is greater than or equal to 0. Then, the ship capsizing probability p in the forecast
condition C can be calculated using the following formula:
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2. Beginning from point xi −1,j (when i = 1, xi −1,j = x0 ), separately calculate the time
and the cost to sail from xi −1,j to xi ,k , (k = 1, 2, . . . , m). The time ti ,j ,k and cost Ci ,j ,k
sailing from xi −1,j to alternative waypoints can be calculated using the following
formula:
Discussion Paper
20
1. Assume that the start time of a ship’s RATC is t0 ; the ship’s position is x0 at t0 , the
next waypoint of ship’s original planned route is xend . Dividing the segment x0 xend
of the route into n parts of equal length, each point is xi (i = 1, 2, . . . , n), and drawing lines perpendicular to the segment x0 xend through points xi (i = 1, 2, . . . , n).
According to the range of the heavy weather area of a TC, m points are chosen on the left semicircular side of the TC for xi (i = 1, 2, . . . , n). These points
xi ,j , (i = 1, 2, . . . , n; j = 1, 2, . . . , m) are the alternative waypoints for a ship’s RATC
(as shown in Fig. 3).
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15
The algorithm of the minimum cost route for RATC is as follows:
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4.2 Algorithm design
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5
waypoints is smaller, the route will be more economical but the computing budget will
increase dramatically. Alternative waypoints can be based on the range of the practical
need and therefore, this extent is a second restrictive condition in the proposed model
for minimum economical cost for RATC. A sketch of alternative waypoints is shown in
Fig. 3.
Practically, in the northern (Southern) Hemisphere, it is much safer to avoid a TC
from the left (right) semicircular side. Therefore in the proposed RATC algorithm, only
waypoints on the safer, left (right) semicircular side of a TC are considered.
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Z
ti ,j ,k =
xi −1,j
1
dx
vship (t, x)
(i = 1, 2, . . . , n; j , k = 1, 2, . . . , m)
(22)
ti ,k
Z
(23)
Taking the ship speed-loss from waves into account, vship (t, x) changes over
time, external environment, and ship course. The real-time change data for wind,
waves, and current can be obtained from the numerical model forecasting results.
Discussion Paper
Coil (D, V , Qa ) + Ct
|
Ci ,j ,k =
ti −1,j
5
4. Calculating the minimum cost CCi ,k = minj (Ci ,j ,k ) and time tt i ,k to reach the point
xi ,k . If CC i ,k = +∞, xi ,k is a broken point that all segments cannot reach, it will
not be used to select the next segment.
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In practice, the fewer the waypoints, the better the route. Fewer waypoints means fewer
changes in direction and subsequent speed loss, resulting in more efficient operation.
Thus, RATC is optimized to reduce the waypoints with no additional cost. The optimization algorithm is illustrated as follows:
|
15
5. Making i = i + 1, and carrying out step two (2) to step five (5) cyclically until i = n.
All the minimum cost segments that connect to the xend (which is xn1 ) make the
shortest time route to avoid TC, which is X0 X1 X2 . . . Xend . The times that a ship
will arrive at each waypoint are T0 , T1 , T2 , . . . , Tn .
Discussion Paper
3. Ascertaining if there is an un-navigable area in a segment of the route xi −1,j xi ,k .
According to the environmental conditions of the ship’s current position, the risk
probability can be assessed in relation to the acceptable risk level. If the risk
probability is acceptable, go to (4). Otherwise, Ci ,j ,k = +∞
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xi ,k
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The TC Nockten was chosen to test the algorithm. Nockten’s track for every six hours
is shown in Fig. 6. Nockten made landfall at Wenchang, Hainan province in China. The
centre of the greatest wind speed is up to 25 m s−1 when landing. The direct economic
loss caused by Nockten is estimated at about 0.6 billion dollars and two people were
killed. Nockten also exerted serious influence on ships at sea.
◦
◦
Assume that a ship was in 19 N, 111.5 E at 00:00 UTC, 28 July 2011, and that
the ship took action to avoid the TC as soon as it received the TC warning. The next
waypoint of the original route was at 20.5◦ N,120◦ E. In this study, this scenario was
used to design the RATC. The ship’s value was 23.8 million dollars while the value of
the cargo was 34.9 million dollars. The price of oil was $871.00 per ton. The ship’s
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5 Case study and results
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4. Let i = z, do step two (2) to step four (4) cyclically until z = n. x ok are the waypoints, and c ok is the cost for the ship as it arrives at waypoints in the optimized
shortest RATC.
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3. Let d c = ccz − cci , and DC = CCz − CCi . If d c ≤ DC, then i = z. Assign xx i and
CC i to set x ok and c ok, respectively. Continue to the next step. If d c ≥ DC, let
z = z − 1 and do step three (3) cyclically.
|
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2. Starting from xx i , calculate the cost ccj occurring when the ship is sailing along
the segment xx i xx j from the xx i to xx j (j = i + 1, i + 2, . . . .). At the same time,
check for un-navigable areas in segment xx i xx j in relation to changing wind and
wave conditions. Let z = j − 1 if there is an un-navigable area.
Discussion Paper
1. The positions of the waypoints, which are the result of the shortest time RATC as
discussed in the last section, are assigned to xx i , (i = 0, 1, . . . , n). The time that
a ship arrives at waypoint xx i is assigned to tt i and the cost to arrive at this point
is CC i . Make i = 0, and continue to the next step;
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5.1 Numerical TC simulation
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To calculate the ship’s capsizing probability, the integration time was set to 0–300 s;
time step was set to 0.0125 s; N was set to 180; The upper limit of the wave power
−1
−1
spectrum was set to 4.5 rad s ; The frequency interval $ was set to 0.025 rad s ; α0
was set to 0.729. The change in the ship’s capsizing probability with changes in the
ship’s heading and wave direction are shown in Fig. 8a. Given the same wave height,
the ship’s capsizing probability reaches a maximum when the beam wave is close.
The ship’s capsizing probability is higher with increasing wave heights (Fig. 8b). The
change in a ship’s capsizing probability with the wave height and the angle between
the heading and wave direction is show in Fig. 8c.
The ship capsizing probability under different wave and heading conditions at 10:00,
28 July is shown in Fig. 9. The figure shows that ship capsizing probability is very
different when the angle between ship’s heading direction and wave direction is different. So, the RATC must consider the angle between ship’s heading direction and wave
Discussion Paper
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5.2 The ship’s capsizing probability
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10
◦
The WRF model was used to simulate the TC. In the model, the 1 ×1 NCEP (National
Center for Environmental Prediction) data for 281 000 was used as the initial data for
the simulation, and six hour time interval daily NCEP data was used as boundary data.
The model simulation area was 99◦ E–130◦ E, 0◦ –30◦ N. The resolution was 0.1◦ × 0.1◦ ,
time step was 300 s. The wind field as forecasted by WRF was used to drive the Wave◦
◦
WatchIII. The resolution of the wave model was 0.1 × 0.1 , time step was 900 s. The
◦
resolution of wave direction was 15 . The results of the two models are shown in Fig. 7.
|
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−1
profitability was 11.1 thousand dollars per day. The fuel consumption was 24 t day , at
a speed of 10 knots (kn). It is assumed that the change of the ship’s displacement was
ignored in the RATC and that the ship’s speed through water was unchanged.
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5.3 Economy and safety experiments
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The ship’s capsizing probability at different heading directions and times were calculated based on the forecast results from the numerical models. The alternative waypoints are shown in Fig. 3, where the interval in the vertical direction of the original
◦
route is 0.35 . The ship speed through water is unchangeable. The experiment of different acceptable capsizing probability and ship speed are shown in Table 3.
To compare the model’s superiority to the sector diagram typhoon avoidance method,
the same experiment is done using sector diagram typhoon avoidance method. In
Fig. 10, A, B, and C separately represent the location of the tropical cyclone centre
at 00:00, 06:00, 12:00 (UTC), on 28 July. H1 , H2 and H3 represent the ship’s location at
00:00, 06:00, 12:00 (UTC), on 28 July. The result of sector diagram typhoon avoidance
method is Exp6. The ship’s experimental RATCs are shown in Fig. 11a. In Fig. 11b, the
ship’s speed is 17 kn and the acceptable capsizing probability of ship is 7 × 10−4 . The
blue line is the RATC before optimization while the red line is the route after optimization. Figure 11b shows that the optimal route can reduce more waypoints. Figure 12
indicates the ship’s position at different times, colours as shown in the capsizing probability scale bar at the bottom of the figure, represent the ship’s capsizing probability at
several heading angles at different times. Because the ship’s speed loss with different
wave directions varies, the shortest route may not be the minimum cost route, which
can be learnt from the experiment results.
Experimental results show that: the higher the acceptable capsizing probability level,
the lower the cost of the RATC. Meanwhile, the higher risk to the ship must be considered. The higher the ship’s speed is, the lower the time cost. However, lower costs do
not mean that the route’s benefit-cost ratio is higher. The case study shows that the cost
of a ship’s RATC is related not only to the ship’s speed and the acceptable risk level,
Discussion Paper
direction. The ship’s capsizing probability is small with stern waves or when the vessel
is sailing head to sea.
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Acknowledgements. This research is supported by the Fundamental Research Funds for the
Central Universities of China, self-determined and innovative research funds of WUT, State
Key Laboratory of Tropical Oceanography (South China Sea Institute of Oceanology, Chinese
Academy of Science).
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A safe-economical RATC was designed based on the dynamic forecast environment.
In this proposed route design method, the limitations of the traditional methods to find
RATC are cancelled or reduced. Numerical forecast errors and the ship’s speed loss
are considered. The ship’s risk under heavy seas is quantified using the ship’s capsizing probability where the ship’s capability to resist wind and waves is considered.
In the proposed model, qualitative judgements inherent to the traditional methods are
avoided. An acceptable risk level should be set according to the ship’s characteristics
and the company’s risk tolerance. This model to avoid TCs not only ensures a ship’s
safety but also reduces shipping costs. The RATC based on the model proposed in this
paper is significantly superior to those produced by traditional methods. Future work
will consider (1) The joint effect of wind and wave will be included when calculating
the ship’s capsizing probability. (2) The geometric growth of computing costs when increasing the precision of alternative waypoints will be addressed (3) The comfort level
of crew.
Discussion Paper
but also to the ship’s performance when resisting wind and waves. Compared to the
Sector diagram typhoon avoidance method, this model is more safe and economical.
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Aertssen, G.: Service Performance and Trials at Sea, Performance Committee of 12th ITTC,
233–265, 1969.
Bedard, R.: Feasibility of Using Wavewatch III for Days-Ahead Output Forecasting for Grid
Connected Wave Energy Projectsin Washington and Oregon, Electric Power, available
at: http://oceanenergy.epri.com/attachments/wave/reports/Wave Forecasting Final Report.
pdf, 78 pp., 2008.
Bikdash, M., Balachandran, B., and Nayfeh, A.: Melnikov analysis for a ship with a general
roll-damping model, Nonlinear Dynam., 6, 101–124, 1994.
Chen, H.: A Minimum Cost Dynamic Program for Ship Routing under Uncertainty, Ph.D. thesis,
Department of Ocean Engineering, MIT, America, 163 pp., 1978.
Chen, H.: Weather Routing: a New Approach, Ocean Systems Inc. USA, available at: http:
//www.ocean-systems.com/pdf docs/Safety%20at%20Sea.pdf (last access: 18 May 2012),
2012.
Chen, J. H. and Zhang, J. P.: Navigation Meteorology and Oceanography, Dalian maritime
university press, Dalian, China, 303 pp., 2001 (in Chinese).
Chu, P. C., Qi, Y. Q., Chen, Y. C., Shi, P., and Mao, Q. W.: South China Sea wind-wave characteristics, Part I: Validation of Wavewatch-III using TOPEX/Poseidon data, J. Atmos. Ocean.
Tech., 21, 1718–1733, 2004.
Delitala, A. M.S, Gallino, S., Villa, L., Lagouvardos, K., and Drago, A.: Weather routing in longdistance Mediterranean routes, Theor. Appl. Climatol., 102, 125–137, 2010.
Falzarano, M., Shaw, S. W., and Troesh, A. W.: Application of global methods for analyzing
dynamic systems to ship rolling motion and capsizing, Int. J. Bifurcat. Chaos, 2, 101–115,
1992.
Gu, J. Y.: Calculation of ship rolling probability using a new path integration method, Journal of
Ship Mechanics, 10, 43–52, 2006.
Guo, Y. Y. and Hou, Y. J.: Statistical analysis of ocean wave numerical forecast error, Marine
Science, 34, 65–68, 2010 (in Chinese).
Hagiwara, H.: Weather routing of (sail-assisted) motor vessels, Ph.D. Thesis, Technical University of Delft, Netherlands, 340 pp., 1989.
He, H. M., Dong, G. X., and Jiang, Y. X.: Approximate estimation for ship speed loss in wave,
Journal of SSSRI, 32, 6–9, 2009.
Discussion Paper
References
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5
Holweg, E. J.: Mariner’s guide for hurricane awareness in the North Atlantic Basin, United
States National Weather Service, available at: http://www.nhc.noaa.gov/marinersguide.pdf
(last access: 18 May 2012), 2000.
Hopkins, J. S.: The Accuracy of Wind and Wave Forecasts, Health and Safety ExecutiveOffshore Techology Report, London, 1997.
Huang, Y. S., Wang, Z., and Li, H. T.: Caclulation of the capsizing probability of ship at beam
wind and sea, Journal of Tianjin University, 34, 651–654, 2001 (in Chinese).
James, R. W.: Application of wave forecast to marime naviagation, US Navy Hydrographic
Office, Washington, 1957.
Li, C., Yang, B., and Zhang, Y. S.: Estimation methods of the ship’s speed-loss in wind and
waves, Ship Science and Technology, 33, 27–30, 2011 (in Chinese).
Liu, D. G., Wang, D. Q., and Wu, Z. L.: Safe-economic decision making model of ship’s avoidance routing to tropical cyclone, Journal of Traffic and Transportation Engineering, 5, 94–98,
2006 (in Chinese).
Padhy, C., Sen, D., and Bhaskaran, P.: Application of wave model for weather routing of ships
in the North Indian Ocean, Nat. Hazards, 44, 373–385, 2008.
Panigrahi, J., Tripathy, J., and Umesh, P.: Optimum tracking of ship routes in 3g-WAM simulated
rough weather using IRS-P4 (MSMR) analysed wind fields, Journal of the Indian Society of
Remote Sensing, 36, 141–158, 2008.
Magirou, E. F. and Psaraftis, H. N.: Quantitative Methods in Shipping: A Survey of Current Use
and Future Trends, Athens university of economics and business, Report No. E115, 1992.
Maki, A., Akimoto, Y., Nagata, Y., Kobayashi, S., Kobayashi, E., Shiotani, S., Ohsawa, T., and
Umeda, N.: A new weather-routing system that accounts for ship stability based on a realcoded genetic algorithm, J. Mar. Sci. Technol., 16, 311–322. doi:10.1007/s00773-011-0128z,2011.
McCord M. R., Young-Kyun Lee, and Hong Kam Lo.: Ship routing through altimetry-derived
ocean currents, Transportation Sci., 33, 49–67, 1999.
Shen, D. and Huang, X. L.: Lasting time before capsize of ship in random beam waves, Shipbuilding of China 41, 14–21, 2000 (in Chinese).
Shi, X.H, Zhang, J., and Wang, S.: Analysis of rolling capsizing probability of warship under
random wind and beam seas, Journal of Ship Mechanics, 15, 473–479, 2011 (in Chinese).
Soda, T., Shiotani, S., Makino, H., and Shimada, Y.: Simulation of weather and ocean for numerical ship navigation, in: Proceedings of the ASME 2011 30th International Conference on
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Ocean, Offshore and Arctic Engineering, Rotterdam, Netherlands, 19–24 June, 2011, 1–8,
2011.
Tang, Y. G., Gu, J. Y., Zheng, H. Y., and Li, H. X.: Study on the ship capsize in random beam
seas using Melnikov method, Journal of Ship Mechanics, 8, 27–34, 2004 (in Chinese).
Thompson, J. M. T.: Transient basins: a new tool for desighing ships against capsize, in: Proceedings, IUTAM Symposium on the Dynamics of Marine Vehicles and Structures in Waves,
London, 325–331, 1990.
Thompson, J. M. T., Rainey, R. C. T., Soliman, M. S.: Mechanics of ship capsize under direct
and parametric wave excitation, Philos. T. R. Soc. Lond., 338, 1–13, 1992.
Tolman, H. L.: Validation of WAVEWATCH III version 1.15 for a global domain,
NOAA/NWS/NCEP/OMB Technical Note Nr. 213, 33 pp., 2002.
Wisniewski, B. and Kaczmarek, P.: Elements of tropical cyclones avoidance procedure,
TransNav – International Journal on Marine Navigation and Safety of Sea Transportation,
6, 119–122, 2012.
Wisniewski, B., Medyna, P., and Chomski, J.: Application of the 1-2-3 rule for calculations of
a Vessel’s route using evolutionary algorithms, TransNav – International Journal on Marine
Navigation and Safety of Sea Transportation, 3, 143–146, 2009.
Wu, J. L., Ma, X. X., Fan, Z. Z., Liu, D. G., and Liu, Z. J.: Benefit evaluation for ship’s avoidance routing of tropical cyclone, Journal of Dalian Maritime University, 36, 31–34, 2010 (in
Chinese).
Zhang, W. J., Wei, S. Y., Li, Q. Liu, D. G., and Liu, Z. J.: Multilevel decision method of ship’s tropical cyclone avoidance route using multisource track forecast, Journal of Traffic and Transportation Engineering, 10, 122–126, 2010 (in Chinese).
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Table 1. The forecast error range division.
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[−2,−1.5]
7.59 × 10−4
[−1.5,−1]
1.6 × 10−2
[−1,−0.5]
0.12
[−0.5,0]
0.34
[0,0.5]
0.36
[0.5,1]
0.14
[1,1.5]
0.02
[1.5,2]
0.001
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Forecast error range (in meters)
Probability in this error range
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I0
C1
C3
φv1
φv2
T
1.070 × 10 kg m
0.871 m
0.013 m
−1.39 rad
1.39 rad
8m
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171.7 m
16.8 m
8000 t
1.070 × 107 kg m2 s−1
1.070 × 107 kg m2
0.807 rad s−1
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B
∆
B1
B2
ω0
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Table 2. Parameter values of the ship.
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No.
Acceptable
capsizing
probability
Pa
Maximum
wind in
the sailing
(m s−1 )
Maximum
wave in the
sailing (m)
Shipping
time (hour)
15
16
17
17
17
17
0.001
0.001
0.001
0.002
0.0007
–
18.1
17.8
20.1
20.1
20.1
20.7
5.2
5.3
4.4
4.4
4.4
4.2
46.6
41.6
41.5
40.7
41.5
40.9
Safety
probability
Cost
(thousand
dollar)
Benefit-cost
ratio
− lg(Ra)
81.37
76.60
79.71
78.17
79.71
78.56
2.858
2.884
2.867
2.875
2.867
2.866
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0.9991
0.9992
0.9991
0.9980
0.9993
0.9959
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Exp1
Exp2
Exp3
Exp4
Exp5
Exp6
Ship
speed
(kn)
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Table 3. Experimental results.
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wave encounter, ωe .
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Fig. 1. The wave encounter angle.
16
Figure 1 The wave encounter angle
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altimeter data in the SCS. The RMS error (σ ) and correlation coefficient between the
6
modelled ( H m ) and observed ( H o ) SWH are 0.48 m and 0.90 respectively. In this research,
7
1330 samples were used for statistical analysis. The distribution of the errors is shown in
8
Figure 2, where ∆H = H m − H o .
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9
Fig.
histogram(a)
of model
errorof
and
(b) scatter
mod- diagram of
10 2. Model
Figureaccuracy
2 Modelstatistics:
accuracy(a)statistics:
histogram
model
error diagram
and (b) of
scatter
elled and observed SWH for all the sample points (Chu et al., 2004).
modelled and observed SWH for all the sample points (Chu et al., 2004)
12
Taking the error range into consideration, a ship capsizing probability for the forecast wave
13
height is calculated based on the distribution of the wave model forecast errors. In this paper,
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the error range was processed using the truncation method. Because the probability of errors
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bigger than 2 m or less than -2 m is only 0.019, this paper only considers the error range in
16
[-2,2]. The division of the error range
1883∆Ei is shown in Table 1, and the probability for each
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Discussion Paper
NHESSD
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Discussion Paper
Safe-economical
route and its
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L. C. Wu et al.
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Fig. 3. Sketch of alternative waypoints.
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Figure 3 Sketch of alternative waypoints
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Practically, in the northern (southern) hemisphere, it is much safer to avoid
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NHESSD
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Discussion Paper
Safe-economical
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assessment model
L. C. Wu et al.
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Figure 4 A route designed to avoid TC
(The break waypoint is the1885
point that all segments cannot reach)
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Fig. 4. A route designed to avoid TC. (The break waypoint is the point that all segments cannot
reach.)
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z
i
z
i
i
i
3
and do step three (3) cyclically.
4
Safe-economical
(4) Let i = z , do step two (2) to step four (4) cyclically until z = n . x _ ok are the waypoints,
route and its
assessment model
NHESSD
and c _ ok is the cost for the ship as it arrives at waypoints in the optimized shortest RATC.
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1, 1857–1893, 2013
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to set x _ ok and c _ ok respectively. Continue to the next step. If dc ≥ DC , let z = z − 1
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Fig.
7 5. Optimized
Figure 5RATC.
Optimized
RATC
10
5. Case study and results.
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The TC Nockten was chosen to test the algorithm. Nockten’s track for every six hours is
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shown in Figure 6. Nockten made landfall at Wenchang, Hainan province in China. The
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1886 is up to 25 m/s when landing. The direct economic loss
centre of the greatest wind speed
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caused by Nockten is estimated at about 0.6 billion dollars and two peopl
Nockten also exerted serious influence on ships at sea.
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1, 1857–1893, 2013
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Safe-economical
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L. C. Wu et al.
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Figure 6 The track of TC Nockten every six hours
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Fig. 6. The track of TC Nockten every six hours.
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of the wave model was 0.1°×0.1°, time step was 900s. The resolution of wave direction was
2
15°. The results of the two models are shown in Figure 7.
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NHESSD
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Safe-economical
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L. C. Wu et al.
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Fig.47. The
forecast
wind and
wave
Figure
7 The forecast
wind
andconditions.
wave conditions
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5.2. The ship’s capsizing probability.
7
To calculate the ship’s capsizing probability, the integration time was set to 0-300s; time step
8
was set to 0.0125s; N was set to 180; The upper limit of the wave power spectrum was set to
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4.5rad/s; The frequency interval ϖ was1888
set to 0.025rad/s; α 0 was set to 0.729. The change
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NHESSD
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Safe-economical
route and its
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L. C. Wu et al.
Title Page
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Fig. 8. The change of ship’s capsizing probability.
Figure 8 The change of ship’s capsizing probability
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The ship capsizing probability under different wave and heading conditions at 1000 28th is
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shown in Figure 9. The Figure shows that ship capsizing probability is very different when the
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angle between ship’s heading direction and wave direction is different. So, the RATC must
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consider the angle between ship’s heading direction and wave direction. The ship’s capsizing
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probability is small with stern waves or when the vessel is sailing head to sea.
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The ship capsizing probability under different wave and heading conditions at 1000 28th is
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shown in Figure 9. The Figure shows that ship capsizing probability is very different when the
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angle between ship’s heading direction and wave direction is different. So, the RATC must
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consider the angle between ship’s heading direction and wave direction. The ship’s capsizing
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probability is small with stern waves or when the vessel is sailing head to sea.
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Discussion Paper
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NHESSD
1, 1857–1893, 2013
Safe-economical
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L. C. Wu et al.
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5.3. Economy and safety experiments.
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The ship’s capsizing probability at different heading directions and times were calculated
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based on the forecast results from the numerical models. The alternative waypoints are shown
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Fig. 9. The ship’s capsizing probability at different ship headings on 28 July 2011, th10:00 (UTC).
11 Figure 9 The ship’s capsizing probability at different ship headings on 1000 28 (UTC)
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and C separately represent the location of the tropical cyclone centre at 280000，280600，
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281200 (UTC). H1 , H 2 and H 3 represent the ship’s location at 280000，280600，281200
NHESSD
8
(UTC). The result of sector diagram typhoon avoidance method is Exp6. The ship’s
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experimental RATCs are shown in Figure 11a.
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1, 1857–1893, 2013
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Safe-economical
route and its
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L. C. Wu et al.
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10Fig. 10. Sector diagram typhoon avoidance method.
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Figure 10 Sector diagram typhoon avoidance method
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Table 3 Experimental results
Acceptable
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indicates the ship’s position at different times, colours as shown in the capsizing probability
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scale bar at the bottom of the figure, represent the ship’s capsizing probability at several
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NHESSD
heading angles at different times. Because the ship’s speed loss with different wave directions
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varies, the shortest route may not be the minimum cost route, which can be learnt from the
9
experiment results.
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Discussion Paper
07
28
12
07
2
80
07
6
28
00
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Fig. 11. (a) Experimental routes (b) The route before and after the optimization experiment
11 Figure 11 (a) Experimental routes (b) The route before and after the optimization
Exp5.
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Exp5
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12
Safe-economical
route and its
assessment model
L. C. Wu et al.
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Introduction
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NHESSD
1, 1857–1893, 2013
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Safe-economical
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Figure 12 Ship position and the capsizing probability at different times
Experimental results show that: the higher the acceptable capsizing probability level, the
5
lower the cost of the RATC. Meanwhile, the higher risk to the ship must be considered. The
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higher the ship’s speed is, the lower the time cost. However, lower costs do not mean that the
1893
route’s benefit-cost ratio is higher. The case study shows that the cost of a ship’s RATC is
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Fig. 12. Ship position and the capsizing probability at different times.
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